is a finite set. Equivalently, \(\mathbf {a}\)  is \(k\) -automatic if there exists a deterministic finite automaton with output (DFAO) that reads the canonical base-\(k\)  representation of \(n\)  and the outputs \(a_n\) . In the case of the PTM sequence, it is clear that \(K_k(\mathbf {t}) =\lbrace \mathbf {t}, -\mathbf {t}\rbrace \) . Equivalently, the sequence is generated by the DFAO in Figure REF . To generate \(t_n\) , one moves between the states (symbolized by nodes) according to subsequent digits in the binary representation of \(n\) . After the final digit has been read, the DFAO returns the output corresponding to the current state. For a detailed treatment of automatic sequences, we refer the reader to the monograph by Allouche and Shallit [1]}.
<FIGURE>